# The polynomial endomorphisms of graph algebras

### Rune Johansen

University of Copenhagen, Denmark### Adam P. W. Sørensen

University of Oslo, Norway### Wojciech Szymański

University of Southern Denmark, Odense, Denmark

## Abstract

We investigate polynomial endomorphisms of graph $C_{∗}$-algebras and Leavitt path algebras. To this end, we define and analyze the coding graph corresponding to each such an endomorphism. We find an if and only if condition for the endomorphism to restrict to an automorphism of the diagonal MASA, which is stated in terms of synchronization of a certain labelling on the coding graph. We show that the dynamics induced this way on the space of infinite paths (the spectrum of the MASA) is generated by an asynchronous transducer.

## Cite this article

Rune Johansen, Adam P. W. Sørensen, Wojciech Szymański, The polynomial endomorphisms of graph algebras. Groups Geom. Dyn. 14 (2020), no. 3, pp. 1043–1075

DOI 10.4171/GGD/574